Semi-algebraic colorings of complete graphs

TL;DR

This paper proves that for multi-colored complete graphs with semi-algebraic color classes of bounded complexity, the Ramsey number grows exponentially with the number of colors, confirming a longstanding conjecture.

Contribution

It establishes an exponential bound on the Ramsey number for semi-algebraic colorings, matching the known lower bounds, and introduces a new regularity lemma for such graphs.

Findings

Proves exponential growth of Ramsey numbers in semi-algebraic colorings.

Develops a Szemerédi-type regularity lemma for multi-colored semi-algebraic graphs.

Confirms a longstanding open problem about the growth rate of Ramsey numbers.

Abstract

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of $m$ is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erd\H{o}s and Shelah.